The idea of a sphere-world was constructed by French mathematician Henri Poincaré who, while pursuing his argument for conventionalism (see philosophy of space and time), offered a thought experiment about a sphere with strange properties.
[1] Poincaré asks us to imagine a sphere of radius R. The temperature of the sphere decreases from its maximum at the center to absolute zero at its extremity such that a body’s temperature at a distance r from the center is proportional to
To finish the story, Poincaré states that the index of refraction will also vary with the distance r, in inverse proportion to
Supposing the inhabitants were to view rods believed to be rigid, or measure distance with light rays.
They would find that a geodesic is not a straight line, and that the ratio of a circle’s circumference to its radius is greater than
This thought experiment is discussed in Roberto Torretti's book Philosophy of Geometry from Riemann to Poincaré[2] and in Jeremy Gray's article "Epistemology of Geometry" in the Stanford Encyclopedia of Philosophy.